Question

\( \frac{1}{\sin^2 \theta} - \frac{1}{\cos^2 \theta} - \frac{1}{\tan^2 \theta} - \frac{1}{\cot^2 \theta} - \frac{1}{\sec^2 \theta} - \frac{1}{\csc^2 \theta} = -3 \), then find the value of \( \theta \).

Hint:

For such problems, use reciprocal and Pythagorean identities effectively to simplify the expressions.

Answer 1

Step 1: Rewrite the given expression using trigonometric identities: \[ \frac{1}{\sin^2 \theta} = \csc^2 \theta, \quad \frac{1}{\cos^2 \theta} = \sec^2 \theta, \quad \frac{1}{\tan^2 \theta} = \cot^2 \theta. \] Substituting these identities: \[ \csc^2 \theta - \sec^2 \theta - \cot^2 \theta - \tan^2 \theta - \sec^2 \theta - \csc^2 \theta = -3. \] Step 2: Simplify: \[ -\sec^2 \theta - \sec^2 \theta - \cot^2 \theta - \tan^2 \theta = -3. \] Step 3: Use the Pythagorean identities: \[ \sec^2 \theta = \tan^2 \theta + 1, \quad \csc^2 \theta = \cot^2 \theta + 1. \] Solving the equation, \( \theta = 45^\circ \).